My somewhat rusty knowledge says that:
$$ \left[A,B\right] = A B - B A =
\left( x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)
\left( x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\right)-
\left( x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}\right)
\left( x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}\right)
$$
The following basic formula from Operator Calculus will be employed repeatedly:
$$
\frac{d}{dx} f(x) = f(x) \frac{d}{dx} + \frac{df}{dx}
$$
Evaluate $(A B)$:
$$
x\frac{\partial}{\partial y} x\frac{\partial}{\partial x} +
x\frac{\partial}{\partial y} y\frac{\partial}{\partial y} -
y\frac{\partial}{\partial x} x\frac{\partial}{\partial x} -
y\frac{\partial}{\partial x} y\frac{\partial}{\partial y} =\\
x\frac{\partial}{\partial y} x\frac{\partial}{\partial x} +
\left[ y\frac{\partial}{\partial y} x\frac{\partial}{\partial y}
+ x\left( \frac{\partial y}{\partial y} \right) \frac{\partial}{\partial y} \right] -
\left[ x\frac{\partial}{\partial x} y\frac{\partial}{\partial x} +
y\left( \frac{\partial x}{\partial x} \right) \frac{\partial}{\partial x} \right] -
y\frac{\partial}{\partial x} y\frac{\partial}{\partial y} =\\
x\frac{\partial}{\partial y} x\frac{\partial}{\partial x} +
y\frac{\partial}{\partial y} x\frac{\partial}{\partial y} +
x\frac{\partial}{\partial y} -
x\frac{\partial}{\partial x} y\frac{\partial}{\partial x} -
y\frac{\partial}{\partial x} -
y\frac{\partial}{\partial x} y\frac{\partial}{\partial y}
$$
Evaluate $(B A)$:
$$
x\frac{\partial}{\partial x} x\frac{\partial}{\partial y} -
x\frac{\partial}{\partial x} y\frac{\partial}{\partial x} +
y\frac{\partial}{\partial y} x\frac{\partial}{\partial y} -
y\frac{\partial}{\partial y} y\frac{\partial}{\partial x} =\\
\left[x\frac{\partial}{\partial y} x\frac{\partial}{\partial x} +
x\left(\frac{\partial x}{\partial x}\right)\frac{\partial}{\partial y}\right] -
x\frac{\partial}{\partial x} y\frac{\partial}{\partial x} +
y\frac{\partial}{\partial y} x\frac{\partial}{\partial y} -
\left[y\frac{\partial}{\partial x} y\frac{\partial}{\partial y} +
y\left(\frac{\partial y}{\partial y}\right)\frac{\partial}{\partial x}\right] =\\
x\frac{\partial}{\partial y} x\frac{\partial}{\partial x} +
x\frac{\partial}{\partial y} -
x\frac{\partial}{\partial x} y\frac{\partial}{\partial x} +
y\frac{\partial}{\partial y} x\frac{\partial}{\partial y} -
y\frac{\partial}{\partial x} y\frac{\partial}{\partial y} -
y\frac{\partial}{\partial x}
$$
We see that $A B = B A$ .
Thus the commutator is zero.
This must be so because rotations and scaling around the origin in the plane commute.
More of the mathematics can be found in: Lie Groups and Variances .